From bd2a63675bb859e42d9e1464ff9a7883884c1fd5 Mon Sep 17 00:00:00 2001
From: Helmut Grohne <helmut@subdivi.de>
Date: Tue, 4 Sep 2012 08:42:38 +0200
Subject: formulate List <-> Vec isomorphism problems

There are now like four open holes in LiftGet.agda that all boil down to
the same problem. For proving equalities on Vec we first need to show
that those Vecs actually have the same type. Proofs on two different
levels are needed and somehow need to be merged.
---
 LiftGet.agda | 30 +++++++++++++++++++++++++++++-
 1 file changed, 29 insertions(+), 1 deletion(-)

diff --git a/LiftGet.agda b/LiftGet.agda
index e2cb653..3c07fd7 100644
--- a/LiftGet.agda
+++ b/LiftGet.agda
@@ -69,6 +69,13 @@ getList-to-getVec get = getlen , get'
         get' : {C : Set} {m : ℕ} → Vec C m → Vec C (getlen m)
         get' v = vec-length (getList-to-getVec-length-property get v) (fromList (get (toList v)))
 
+{-
+-- We cannot formulate the first commutation property, because the type of
+-- fromList (get l) depends on the concrete l, more specifically its length.
+get-commut-1 : (get : get-type) → (fromList ∘ get) ≗ (proj₂ (getList-to-getVec get)) ∘ fromList
+get-commut-1 get l = ?
+-}
+
 get-trafo-1 : (get : get-type) → {B : Set} → getVec-to-getList (proj₂ (getList-to-getVec get)) {B} ≗ get {B}
 get-trafo-1 get l = begin
   getVec-to-getList (proj₂ (getList-to-getVec get)) l
@@ -92,6 +99,18 @@ length-replicate : {A : Set} {a : A} → (n : ℕ) → length (replicate n a) 
 length-replicate 0       = refl
 length-replicate (suc n) = cong suc (length-replicate n)
 
+{-
+-- We cannot write the following property, because the expressions
+-- fromList (toList v) and v have different type. The type differs in the
+-- length. Somehow we would have to embed a proof that those types are in fact
+-- the same into the type signature of the function.
+fromList-toList : {A : Set} {n : ℕ} → (v : Vec A n) → fromList (toList v) ≡ v
+fromList-toList v = ?
+-}
+
+get-commut-2 : (getlen : ℕ → ℕ) → (get : getV-type getlen) → {B : Set} {n : ℕ} → (toList ∘ get {B} {n}) ≗ (getVec-to-getList get) ∘ toList
+get-commut-2 getlen get v = {!!}
+
 get-trafo-2-getlen : (getlen : ℕ → ℕ) → (get : getV-type getlen) → proj₁ (getList-to-getVec (getVec-to-getList get)) ≗ getlen
 get-trafo-2-getlen getlen get n = begin
   proj₁ (getList-to-getVec (getVec-to-getList get)) n
@@ -106,4 +125,13 @@ getVec-getlen : {getlen₁ getlen₂ : ℕ → ℕ} → (get : getV-type getlen
 getVec-getlen get p {B} {n} v = vec-length (p n) (get v)
 
 get-trafo-2-get : (getlen : ℕ → ℕ) → (get : getV-type getlen) → {B : Set} {n : ℕ} → proj₂ (getList-to-getVec (getVec-to-getList get)) {B} {n} ≗ getVec-getlen get (sym ∘ (get-trafo-2-getlen getlen get))
-get-trafo-2-get getlen get v = {!!}
+get-trafo-2-get getlen get v = begin
+  proj₂ (getList-to-getVec (getVec-to-getList get)) v
+    ≡⟨ refl ⟩
+  vec-length (getList-to-getVec-length-property (getVec-to-getList get) v) (fromList (toList (get (fromList (toList v)))))
+    ≡⟨ {!!} ⟩
+  vec-length (sym (get-trafo-2-getlen getlen get (vec-len v))) (vec-length (cong getlen (length-toList v)) (get (fromList (toList v))))
+    ≡⟨ {!!} ⟩
+  vec-length (sym (get-trafo-2-getlen getlen get (vec-len v))) (get v)
+    ≡⟨ refl ⟩
+  getVec-getlen get (sym ∘ (get-trafo-2-getlen getlen get)) v ∎
-- 
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